On the compatibility of algebraic equations with extensions

Medientyp: E-Artikel
Titel: On the compatibility of algebraic equations with extensions
Beteiligte: Hule, Harald; Müller, Winfried B.
Quelle: Journal of the Australian Mathematical Society ; 21 ( 1976 ) S. 381-383
Erschienen: Cambridge University Press (CUP), 1976
Sprache: Englisch
DOI: 10.1017/s1446788700018693
ISSN: 1446-7887; 1446-8107
Schlagwörter: General Mathematics
Zusammenfassung: <jats:title>Abstract</jats:title><jats:p>In this paper we investigate the following problem proposed by Lausch and Nöbauer: Let <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S1446788700018693_inline1" /> be a variety of universal algebras, <jats:italic>B</jats:italic> an algebra of <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S1446788700018693_inline1" /> and <jats:italic>A</jats:italic> a subalgebra of <jats:italic>B</jats:italic>. If a system of algebraic equations over <jats:italic>A</jats:italic> is solvable in <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S1446788700018693_inline1" />, is it then solvable over <jats:italic>B</jats:italic>? We show that the answer is affirmative in certain varieties but negative in the general case.</jats:p>
Beschreibung: <jats:title>Abstract</jats:title><jats:p>In this paper we investigate the following problem proposed by Lausch and Nöbauer: Let <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S1446788700018693_inline1" /> be a variety of universal algebras, <jats:italic>B</jats:italic> an algebra of <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S1446788700018693_inline1" /> and <jats:italic>A</jats:italic> a subalgebra of <jats:italic>B</jats:italic>. If a system of algebraic equations over <jats:italic>A</jats:italic> is solvable in <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S1446788700018693_inline1" />, is it then solvable over <jats:italic>B</jats:italic>? We show that the answer is affirmative in certain varieties but negative in the general case.</jats:p>

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